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\(S\)-integral points on elliptic curves – notes on a paper of B. M. M. de Weger. (English) Zbl 1065.11014
B. M. M. de Weger solved the Diophantine equation \(y^2=x^3-228x+848\) (*) [J. Théor. Nombres Bordeaux 9, 281–301 (1997; Zbl 0898.11008)] in \(S\)-integers with \(S=\{2,\infty\}\), by using tools from Algebraic Number Theory and lower estimates for linear forms in complex and \(q\)-adic logarithms of algebraic numbers. In the present paper a shorter solution for the \(S\)-solutions to (*) is given, for a larger set \(S\), namely \(S=\{2,3,5,7,\infty\}\). Now, linear forms in elliptic logarithms are used and the basic tool for computing lower bounds for such linear forms is a theorem of G. Rémond and F. Urfels [J. Number Theory 57, No. 1, 133–160 (1996; Zbl 0853.11055)], which applies to elliptic curves of rank at most 2, hence to the elliptic curve defined by (*). It should be noted that the same authors in cooperation with J. Gebel and H. G. Zimmer have given an alternative approach to (*), which avoids linear forms in \(q\)-adic elliptic logarithms [Math. Proc. Camb. Philos. Soc. 127, No. 3, 383–402 (1999; Zbl 0949.11033)], but the bounds resulting there are much larger than those of the present paper which result from the theorem of Rémond and Urfels.
MSC:
11D25 Cubic and quartic Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
11G05 Elliptic curves over global fields
Software:
Magma; SIMATH
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References:
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